Friday, December 19, 2014

The deltahedra inside voxel-based solids

There's a tetrahedron inside every cube (and in two possible orientations.) (Image quoted from http://burrtools.sourceforge.net/gui-doc/Spacegrids.html )

There is a tetrahedron inside every cube or voxel: its edges are six face diagonals. There are two possible orientations for the tetrahedron as there are two consistent sets of face diagonals. If every unit-cube voxel of Cartesian coordinates (x, y, z) is assigned the parity of x+y+z, and tetrahedron orientations are assigned according to this parity, neighboring tetrahedra always meet edge to edge.

In this way a vox solid (voxel-based solid) can be converted into a deltahedron (a solid faceted exclusively with equilateral triangles) that lies entirely within the original vox solid.

A structural problem arises with this technique. For example, a rigid stack of voxels becomes a flexible chain of tetrahedra.

The tetrahedra that lie inside a rigid stack of cubes form a flexible chain. (Image quoted from http://www.kaleidocycles.de )

A solution to this structural problem is to divide each voxel into eight sub-voxels, and replace those sub-voxels by tetrahedra. Here again there is a parity choice, one that makes these eight tetrahedra either a stella octangula or a "cumulated cuboctahedron."

A "cumulated cuboctahedron" (shown in red) inserted between the two halves of a stella octangula (gray.) (Image quoted from http://the-arc-ddeden.blogspot.com )

If stella octangula voxels are used, the corners are prominent. Using cumulated cuboctahedra (which can be imagined by slicing a half-voxel margin off this object) results in bevelled corners.

Whether the stella octangula or "cumulated cuboctahedra" units are chosen, weaving their surface triangles in corrugated kagome (a.k.a., knotology) restores the missing cube corners of the sub-cubes—the woven piece resembles the original vox solid externally, but its internal structure is a little different.